## Isothermal Expansion of an Ideal Gas

An isothermal expansion of a gas is such that the temperature of the gas is kept constant. The diagram below illustrates the isothermal expansion of a gas fromtoduring which the pressure falls fromto

The gas expands during this expansion, so work is done by the gas. The amount of work done is equal to the area under the graph. We can write it as an integral:

If the gas is ideal then we can sue the ideal gas equationrearranged asto obtain the integral

We can evaluate this integral quite easily, obtaining(1)

Becausethe gas does work so W is positive.

Notice also thatfor an isothermal process so thatThis means we can write equation (1) as